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12.08.2020 •
Mathematics
Find the first six partial sums S1, S2, S3, S4, S5, S6 of the sequence whose nth term is given. 1 2 , 1 22 , 1 23 , 1 24 , . .
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Ответ:
the first partial sum![\mathbf{S_1= \dfrac{1}{2}}](/tpl/images/0718/4971/15a4d.png)
the second partial sum![\mathbf{S_2= \dfrac{3}{4} }](/tpl/images/0718/4971/ed42e.png)
the third partial sum![\mathbf{S_3= \dfrac{7}{8}}](/tpl/images/0718/4971/1f192.png)
the fourth partial sum![\mathbf{S_4= \dfrac{15}{16}}](/tpl/images/0718/4971/65d20.png)
the fifth partial sum![\mathbf{S_5= \dfrac{31}{32}}](/tpl/images/0718/4971/e99d6.png)
the sixth partial sum![\mathbf{S_6= \dfrac{63}{64}}](/tpl/images/0718/4971/2600e.png)
Step-by-step explanation:
The term of the sequence are given as :
,
,
,
, . . .
The nth term for this sequence is ,![\mathtt{a_n =( \dfrac{1}{2})^n}](/tpl/images/0718/4971/da1e9.png)
The nth partial sum of the sequence for
is ![\mathtt{S_n}](/tpl/images/0718/4971/9fe6f.png)
where;
The first partial sum is:![\mathtt{S_1= a_1}](/tpl/images/0718/4971/492a7.png)
Therefore, the first partial sum![\mathbf{S_1= \dfrac{1}{2}}](/tpl/images/0718/4971/15a4d.png)
The second partial sum is:![\mathtt{S_2= a_1+a_2}](/tpl/images/0718/4971/7f7a7.png)
Therefore, the second partial sum![\mathbf{S_2= \dfrac{3}{4} }](/tpl/images/0718/4971/ed42e.png)
The third partial sum is :![\mathtt{S_3= a_1+a_2+a_3}](/tpl/images/0718/4971/25e40.png)
Therefore, the third partial sum![\mathbf{S_3= \dfrac{7}{8}}](/tpl/images/0718/4971/1f192.png)
The fourth partial sum :![\mathtt{S_4= a_1+a_2+a_3+a_4}](/tpl/images/0718/4971/a7111.png)
Therefore, the fourth partial sum![\mathbf{S_4= \dfrac{15}{16}}](/tpl/images/0718/4971/65d20.png)
The fifth partial sum :![\mathtt{S_5= a_1+a_2+a_3+a_4+a_5}](/tpl/images/0718/4971/8033a.png)
Therefore, the fifth partial sum![\mathbf{S_5= \dfrac{31}{32}}](/tpl/images/0718/4971/e99d6.png)
The sixth partial sum:![\mathtt{S_5= a_1+a_2+a_3+a_4+a_5+a_6}](/tpl/images/0718/4971/10d06.png)
Therefore, the sixth partial sum![\mathbf{S_6= \dfrac{63}{64}}](/tpl/images/0718/4971/2600e.png)
Ответ:
she would have to work 8 hours